The Quantum Structure of Space and Time (293 pages)

172 The Quantum Structure of Space and Tame

a CFT on a 3-sphere, this phase transition is the Hawking-Page transition, and the

dual high temperature background is Ads-Schwarzschild. Both above and below
the transition the bulk asymptotes to (nearly) Ads. Most of it remains cold and it
is not sensitive to the short distance behavior of string theory.
While the boundary field theory is manifestly local, locality in the bulk is subtle.
Because of the infinite warp factor, possible violation of locality in the bulk over

distances of order 1, could be consistent with locality at the boundary. In fact, it is

quite difficult to find operators in the field theory which represent events in the bulk
which are localized on scales of order RAds or smaller. This underscores the fact
that it is not clear what we mean by locality, if all we can measure are observables
at infinity.
These developments have led to many new insights about the two sides of the
duality and the relation between them (for a review, see [21]). In particular, many
new results about gauge theories, including their strong coupling phenomena like
thermal phase transitions, confinement and chiral symmetry breaking were eluci-
dated. The main new insight about gravity is its holographic nature - the boundary
theory contains all the information about the bulk gravity theory which is higher

dimensional. Therefore, the number of degrees of freedom of a gravity theory is not

extensive. This is consistent with the lack of local observables in gravity.

5.1.5.4

Generalities Another class of examples of an emergent space dimension involves

backgrounds with a linear dilaton direction. The string coupling constant depends

on the position in the emergent direction, parameterized by the spatial coordinate

4, through gs(4) = eT with an appropriate constant Q. Therefore, the string

coupling constant vanishes at the boundary 4 -+ -co. The other end of the space

at 6 4 +m is effectively compact.

Like the Ads examples, here the bulk string theory is also dual to a theory

without gravity at the boundary. In that sense, this is another example of hologra-

phy. However, there are a few important differences between this duality and the

AdS/CFT duality.

In most of the linear dilaton examples the holographic theory is not a standard

local quantum field theory. For example, the near horizon geometry of a stack of

NS5-branes is a linear dilaton background which is holographic to the little string

theory (for a review, see e.g. Ill]). The stringy, non-field theoretic nature of the

holographic theory follows from the fact that it has nonzero a’, and therefore it

exhibits T-duality.

Because of the vanishing interactions at the boundary of the space, the in-

teractions take place in an effectively compact region (the strong coupling end).

Therefore, we can study the S-matrix elements of the bulk theory. These are the

observables of the boundary theory.

Unlike the Ads examples, the string metric does not have an infinite warp factor.

Emergent space with gravity: linear dilaton backgrounds

Qd